Cubic group representation

Each of these configurations, except d1 and d9, will give rise to terms under the action of HER as a perturbation. The terms, of course, bear the labels of the cubic group, here Oh. The terms which arise are determined qualitatively by the decomposition of the group theoretical direct product of the electrons (or holes) involved into irreducible representations of the cubic groups. A t2s or eR orbital set more than half-full is treated as the equivalent number of holes, and a filled one is ignored. Then, for instance... 

The application of the reduction formula is exemplified by the decomposition of the D2 representation of the group R3 (Table 65) in terms of the irreducible representations of the cubic group O having the character table according to Table 64. Now the appearances of the individual irreducible representations are evaluated according to the reduction formula of the form... 

To generate an irreducible G subspace, for particular cases, f needs to be chosen with care. In the case of the kubic harmonics, first defined by Bethe in 1929 suitable functions are the mononomials x y"zP, which we identify in Elert s notation as (mnp). The kubic harmonics up to level 4 and their maps onto the irreducible representations of the cubic groups are listed in Table 3.9. 

The index v in Eq. (85c) occurs because of the fact that in any point group there are three components of the hyperpolarizability that belong to the same representation as the components of the vector p. In particular, for cubic groups these three sets are... 

The parity of D > may be deduced from the fact that the d-orbitals, which are basis functions for D > contain the spherical harmonics which do not change sign upon reflection in the origin. Therefore the representation D ) is of even parity. This property is carried over into the cubic group so that we may now say that D > has been reduced to the representations eg and t2g in the group Oh, where the suffix g indicates even parity. 

That is, for any q-group, the representations fi = 0 and fj, = q/2 are nondegenerate and all others are doubly degenerate. This notation clearly does not cover the case of cubic groups, in which triplets and quartets occur. Also, if the group contains rotations perpendicular to the z-axis or reflections through planes containing the z-axis, a superscript (-I-) or (-) is necessary to uniquely specify the one-dimensional representations. Allowed values of q are q = 1, 2, 3, 4, or 6. 

Invariance under the point-group operations requires that the crystal-field Hamiltonian contain only operators that transform as the identity representation of the point group. These operators are easy to determine in general, since, for all the point groups except the cubic groups (T, Tj, T, O, and Oh), all group operators may be constructed from the following operators (Leavitt, 1980) ... 

The three components of the rotational tensor linear order. These symmetry strains are shown in fig. 4. p = corresponds to the fully symmetric volume strain. Deformations of the local environment lead to deformations of the 4f-charge cloud, microscopically one therefore has a coupling of strains to multipolar operators Op of the 4f shell. These are polynomials in J, and of degree 1 = 2, 4 and 6 which again transform as irreducible point-group representations. In the cubic case the quadrupolar (/ = 2) operators are ... 

The complete Character Table of D4/1 can be found in Appendix A. A point worth noting is that the number of irreducible representations, here 10, is the same as the number of classes of sym-ops in the group. The reader might care to check this point in the Character Tables of some of the other listed symmetry point groups, such as the cubic groups, and O/, which have three-dimensional irreps (labelled T) because the x, y and z axes are energetically equivalent, i.e. triply degenerate, in them. 

Let us now consider the necessary conditions for the appearance of phonons in impurity-ion electronic spectra. The presence of a substitutional defect in an otherwise perfect crystal removes the translational symmetry of the system and reduces the symmetry group of the system from the crystal space group to the point group of the lattice site. Loudon [26] has provided a table for the reduction of the space group representations of a face-centered cubic lattice into a sum of cubic point-group representations. A portion of that table is shown in Table 1 here. Consider an impurity ion that undergoes a vibronic electric-dipole allowed transition, with T and Tf the irreducible representations of the initial and final electronic states. Since the electric dipole operator transforms as Tj in the cubic point group, Oh, the selection rule for participation of a phonon is that one of its site symmetry irreducible representations is contained in the direct product T x Ti X Tf. 

For cubic groups one has to work with irreducible representations. The crystal quantum numbers are not useful to classify the states, because of the occurrence of triplet and quartet states. Quartet states are found in the double groups OJ, O and Tj. 

However, for problems involving cubic symmetry, the functions given in Table AIV.l are awkward to use since they do not directly form triply degenerate sets, despite the fact that the entire set of / functions spans the representations At, TXu, and T in the group Oh. 

Here x(C4), for instance, means the character for the covering operation consisting of rotation about one of the fourfold or principal cubic axes (normals to cube faces) by 2a-/4. Any rotation about such an axis leaves two atoms invariant, and hence x(Cs) = x(C4) = 2- On the other hand, x((Y)=x(Q)—0 since no atoms are left invariant under rotations about the twofold or secondary cubic axes (surface diagonals) or about the threefold axes (body diagonals). Inversion in the center of symmetry is denoted by I. By using tables of characters for the group Oh, one finds that the irreducible representations contained in the character scheme (2) are, in MuIIiken s notation,4... 

The following operators were used in Chapter 7 as representative operators of the five classes of the cubic point group O E, R(2n/3 [1 1 1]), R(n/2 z), R(n z), R(n [1 1 0]). Derive the standard representation for these operators and show that this representation is irreducible. [Hint You may check your results by referring to the tables given by Altmann and Herzig (1994) or Onadera and Okasaki (1966).]... 

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